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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeDec 28th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea Right Bousfield localizations of [[model categories]] are a way to encode [[coreflective localizations]] of [[(∞,1)-categories]] in the setting of [[model categories]]. Given a [[model category]] $M$, the formal definition and elementary properties can be deduced from the definition and properties of [[left Bousfield localizations]] of the [[opposite category]] of $M$. However, the typical existence theorems for right Bousfield localizations are not formulated in this manner, since typical conditions involve some form of [[local presentability]] of the underlying category, or an analogous condition, which typically do not hold for the [[opposite category]]. ## Barwick's existence theorem If $C$ is a [[tractable model category]], $D$ is an accessible accessibly embedded subcategory of $C$ that is closed under [[homotopy colimits]] in $C$. Then there is a right Bousfield localization of $C$, which is a [[right model category]] (but not necessarily a [[model category]]), and its colocal objects are precisely the object of $D$. ## Related concepts * [[left Bousfield localization of model categories]] ## References The original article: * {#Bousfield75} [[A. K. Bousfield]], *The localization of spaces with respect to homology*, Topology **14** (1975), 133–150 &lbrack;<a href="http://dx.doi.org/10.1016/0040-9383(75)90023-3">doi:10.1016/0040-9383(75)90023-3</a>&rbrack; Detailed discussion (including existence results for [[left proper model category|left proper]] [[cellular model category|cellular model categories]]): * {#Hirschhorn02} [[Philip Hirschhorn]], Chapter 5 of: _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs 99 (2002) &lbrack;[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)&rbrack; Existence results for [[combinatorial model categories]]: * {#Barwick10} [[Clark Barwick]], *On left and right model categories and left and right Bousfield localizations*, Homology, Homotopy and Applications **12** 2 (2010) 245–320 &lbrack;[doi:10.4310/hha.2010.v12.n2.a9](https://doi.org/10.4310/hha.2010.v12.n2.a9), [euclid:1296223884](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/On-left-and-right-model-categories-and-left-and-right/hha/1296223884.full), subsuming: [arXiv:0708.2067](https://arxiv.org/abs/0708.2067), [arXiv:0708.2832](https://arxiv.org/abs/0708.2832), [arXiv:0708.3435](http://arxiv.org/abs/0708.3435)&rbrack; Right Bousfield localization specifically for [[stable model categories]] (such as [[model structures on spectra]]): * D. Barnes and C. Roitzheim. Stable left and right Bousfield localisations. Glasg. Math. J., 56(1):13–42, 2014. [[!redirects right Bousfield localization]] <a href="https://ncatlab.org/nlab/revision/right+Bousfield+localization+of+model+categories/1">v1</a>, <a href="https://ncatlab.org/nlab/show/right+Bousfield+localization+of+model+categories">current</a>

   Created:

   Idea

   Right Bousfield localizations of model categories are a way to encode coreflective localizations of (∞,1)-categories in the setting of model categories.

   Given a model category $M$, the formal definition and elementary properties can be deduced from the definition and properties of left Bousfield localizations of the opposite category of $M$.

   However, the typical existence theorems for right Bousfield localizations are not formulated in this manner, since typical conditions involve some form of local presentability of the underlying category, or an analogous condition, which typically do not hold for the opposite category.

   Barwick’s existence theorem

   If $C$ is a tractable model category, $D$ is an accessible accessibly embedded subcategory of $C$ that is closed under homotopy colimits in $C$. Then there is a right Bousfield localization of $C$, which is a right model category (but not necessarily a model category), and its colocal objects are precisely the object of $D$.

   Related concepts

   • left Bousfield localization of model categories

   References

   The original article:

   • {#Bousfield75} A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150 [doi:10.1016/0040-9383(75)90023-3]

   Detailed discussion (including existence results for left proper cellular model categories):

   • {#Hirschhorn02} Philip Hirschhorn, Chapter 5 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf]

   Existence results for combinatorial model categories:

   • {#Barwick10} Clark Barwick, On left and right model categories and left and right Bousfield localizations, Homology, Homotopy and Applications 12 2 (2010) 245–320 [doi:10.4310/hha.2010.v12.n2.a9, euclid:1296223884, subsuming: arXiv:0708.2067, arXiv:0708.2832, arXiv:0708.3435]

   Right Bousfield localization specifically for stable model categories (such as model structures on spectra):

   • D. Barnes and C. Roitzheim. Stable left and right Bousfield localisations. Glasg. Math. J., 56(1):13–42, 2014.

   !redirects right Bousfield localization

   v1, current